Hall conductance of a non-Hermitian two-band system with k-dependent decay rates
HHall conductance of a non-Hermitian two-band system with k − dependent decay rates Junjie Wang, Fude Li, and X. X. Yi
1, 2, ∗ Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun 130024, China Center for Advanced Optoelectronic Functional Materials Research,and Key Laboratory for UV-Emitting Materials and Technology of Ministry of Education,Northeast Normal University, Changchun 130024, China (Dated: January 8, 2021)Two-band model works well for Hall effect in topological insulators. It turns out to be non-Hermitian whenthe system is subjected to environments, and its topology characterized by Chern numbers has received exten-sive studies in the past decades. However, how a non-Hermitian system responses to an electric field and whatis the connection of the response to the Chern number defined via the non-Hermitian Hamiltonian remain barelyexplored. In this paper, focusing on a k − dependent decay rate, we address this issue by studying the responseof such a non-Hermitian Chern insulator to an external electric field. To this aim, we first derive an effectivenon-Hermitian Hamiltonian to describe the system and give a specific form of k − dependent decay rate. Thenwe calculate the response of the non-Hermitian system to a constant electric field. We observe that the envi-ronment leads the Hall conductance to be a weighted integration of curvature of the ground band and hencethe conductance is no longer quantized in general. And the environment induces a delay in the response of thesystem to the electric field. A discussion on the validity of the non-Hermitian model compared with the masterequation description is also presented. I. INTRODUCTION
Since the discovery of quantum Hall effect in the 1980s,the topological band theory has been extensively developedand applied in various systems, ranging from insulators andsemimetals to superconductors [1–3]. In the theory, the Chernnumber captures the winding of the eigenstates and is de-fined via the integral of the Berry curvature over the first Bril-louin zone. It can not only be used to classify topologicalmaterials, but also quantify the response of the system to anexternal field. For example, the Hall conductance is quan-tized and is proportional to the sum of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) invariants(or Chern numbers)of all filled bands [4–6]. This theory was previously estab-lished in closed systems, and one may wonder if it holds validfor open systems.Open systems can be described effectively by non-Hermitian Hamiltonian. Recently, non-Hermitian topologi-cal theories have been introduced and experiments have beenconducted in dozens of open systems [7–32]. Many inter-esting features are predicted and observed in non-Hermitiansystems [10–13], including the band defined on the com-plex plane [10], the breakdown of the conventional bulk-boundary correspondence [11–13] and the non-Hermitian skineffect [11]. The definition of the topological invariance fornon-Hermitian systems has also been discussed [10, 20].From the side of response, the impact of non-Herminityon observable has been studied using field-theoretical tech-niques within the linear response theory(e.g. the Kubo for-mula) [21–25]. They found that there is no link between non-Hermitian topological invariants and the quantization of ob-servables [22] and the observables are no longer quantizedin general otherwise requiring more strict conditions than a ∗ Electronic address: [email protected] nonzero non-Hermitian Chern number [23]. In these studies,the non-Hermitian term was introduced via self-energy in thelow-energy limit [21], or phenomenologically introduced toadd into the system, so a non-Hermitian version of the TKNNcan be derived to show the topological contribution [23]. Thisgives rise a question that how the decay rate depends on themomentum of the electron, and how the decay depends on thecouplings between the system and the environment, and how anon-Hermitian system with k − dependent decay rate responseto an external stimulus.In this work, we will answer these questions and shed morelight on the response theory for non-Hermitian two-band sys-tems by the adiabatic perturbation theory [33–35]. The re-minder of this paper is organized as follows. In Sec. II, we in-troduce the system-environment couplings and derive a masterequation by which we obtain a non-Hermitian Hamiltonian todescribe the two-band system subjected to environments. Thedependence of the decay rates on the Bloch vectors is also de-rived and discussed. In Sec. III, we work out the response ofnon-Hermitian two-band systems to a constant electric fieldusing the adiabatic perturbation theory. The results show thatthe response of the system can be divided into two terms. Thefirst term is proportional the non-Hermitian Chern number fortwo-band systems, which is reminiscent of the relation be-tween the Chern number and the Hall conductance of closedsystems. While the second term can be treated as a correctionthat suggests the relationship between the Chern number andthe conductance might fail for open systems. In Sec. IV, tak-ing a tight-binding electron in a two-dimensional lattice as anexample, we calculate and plot the complex-band structuresas a function of the momentum of the electron, k x and k y .The Hall conductance of the system is also shown and ana-lyzed. We find that Hall conductance depends on the strengthof the decay rate and the electron occupation on the Blochband, which no longer leads to a quantized Hall conductanceand a delay in the response of the system to the constant elec-tric field appears. We also compare the Hall conductance of a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n the open systems by non-Hermitian Hamiltonian and by mas-ter equation in this section. In Sec. V, we conclude the paperwith discussions. II. MASTER EQUATION AND EFFECTIVENON-HERMITIAN HAMILTONIAN
We start with a two-band Chern insulator, whose Hamilto-nian can be expressed as (setting (cid:126) = 1 ) ˆ H S ( (cid:126)k ) = d ( (cid:126)k ) σ + (cid:126)d ( (cid:126)k ) · (cid:126)σ, (1)where σ is the × identity matrix, (cid:126)σ = ( σ x , σ y , σ z ) arePauli matrices and (cid:126)k = ( k x , k y ) represent the Bloch vectorof the electron. d ( (cid:126)k ) is a shift of zero energy level and canbe ignored [36] for simplicity. (cid:126)d ( (cid:126)k ) = ( d x ( (cid:126)k ) , d y ( (cid:126)k ) , d z ( (cid:126)k )) are three component vectors depending on the feature of ma-terials under research. The eigenvalues of this Hamiltonianare (cid:15) ± ( k ) = ± d ( (cid:126)k ) , with d ( (cid:126)k ) = (cid:113)(cid:80) i = x,y,z d i ( (cid:126)k ) . Thecorresponding eigenstates of the model read, | k + (cid:105) = cos( θ e − iφ | + (cid:105) + sin( θ |−(cid:105) , | k −(cid:105) = sin( θ e − iφ |−(cid:105) − cos( θ |−(cid:105) , (2)where the parameter θ = arccos( d z ( (cid:126)k ) /d ( (cid:126)k )) ,φ = arctan( d y ( (cid:126)k ) /d x ( (cid:126)k )) . (3)Considering an environment as quantized electromagneticfield of many modes coupled to the system, by substitution (cid:126)k → ( (cid:126)k − e (cid:126)A ) , where the elementary charge of electron is − e ( e > ), (cid:126)A = − (cid:80) nj (cid:126)e j g n ( b n + b † n ) with (cid:126)e j standing for theunit vector, g n = (cid:112) / V (cid:15) ω n and V being the volume of theenvironment, the Hamiltonian up to the first order in (cid:126)A reads, ˆ H ( (cid:126)k ) = ˆ H S ( (cid:126)k ) + ˆ H R + ˆ H I ( (cid:126)k ) ≈ (cid:126)d ( (cid:126)k ) · (cid:126)σ + (cid:88) n ω n ˆ b † n ˆ b n − e (cid:88) i = x,y,z ( (cid:79) (cid:126)k d i · (cid:126)A ) σ i , (4)where ω n is the eigenfrequency of the environment and (cid:15) is the vacuum permittivity. ˆ b † n and ˆ b n are creation and an-nihilation operators of the nth mode of the environment, re-spectively. (cid:79) (cid:126)k stands for the partial derivative with respectto (cid:126)k . For simplicity, we set (cid:79) k ( t ) = (cid:79) k . Simple algebrawith rotating-wave approximations [37] shows that the totalHamiltonian in the interaction picture takes ˆ H I ( t, (cid:126)k ) = (cid:88) n g n ( (cid:126)k ) σ + ˆ b n e i ( d ( (cid:126)k ) − ω n ) t + H . c , (5) where g n ( (cid:126)k ) = g ( ω n ) D , g ( ω n ) = − eg n , σ + ≡ | k + (cid:105)(cid:104) k − | , σ − ≡ | k −(cid:105)(cid:104) k + | and D + = (cid:79) (cid:126)k d x ( (cid:126)k )(cos φ cos θ + i sin φ )+ (cid:79) (cid:126)k d y ( (cid:126)k )(sin φ cos θ − i cos φ ) − (cid:79) (cid:126)k d z ( (cid:126)k ) sin θ,D − ≡ D ∗ + . (6)We assume that the initial states of the system and environ-ment are separable, i.e., ρ ISR (0) = ρ S (0) ⊗ ρ R (0) . In theBorn-Markov approximaton, we obtain the following masterequation for the system [38, 39] ˙ ρ S ( t ) = − i (cid:126) (cid:104) ˆ H S ( (cid:126)k ) , ρ S (cid:105) + U ( ρ S ) + D ( ρ S ) , (7)where U ( ρ S ) = γ ( (cid:126)k )( N ( (cid:126)k ) + 1) (cid:104) σ − ( (cid:126)k ) ρ S σ + ( (cid:126)k ) − σ + ( (cid:126)k ) σ − ( (cid:126)k ) ρ S − ρ S σ + ( (cid:126)k ) σ − ( (cid:126)k ) (cid:105) , D ( ρ S ) = γ ( (cid:126)k ) N ( (cid:126)k ) (cid:104) σ + ( (cid:126)k ) ρ S σ − ( (cid:126)k ) − σ − ( (cid:126)k ) σ + ( (cid:126)k ) ρ S − ρ S σ − ( (cid:126)k ) σ + ( (cid:126)k ) (cid:105) . (8)Here, γ ( (cid:126)k ) is the decay rate that depends on the Bloch vectors. N ( v n ) = { exp [ ω n /k B T ] − } − , represents the averagenumber of photons in the system. In the Weisskopf-Wignerapproximation, N ( (cid:126)k ) = (cid:110) exp (cid:104) d ( (cid:126)k ) /k B T (cid:105) − (cid:111) − . In thelimit T → , we find that N ( (cid:126)k ) → and D ( ρ S ) → . Withthis consideration, the master equation can be written as ˙ ρ S ( t ) = − i (cid:104) ˆ H S ( (cid:126)k ) , ρ S (cid:105) + γ ( (cid:126)k ) (cid:104) σ + ( (cid:126)k ) ρ S σ − ( (cid:126)k ) − σ − ( (cid:126)k ) σ + ( (cid:126)k ) ρ S − ρ S σ − ( (cid:126)k ) σ + ( (cid:126)k ) (cid:105) = − i (cid:16) ˆ H eff ρ S − ρ S ˆ H † eff (cid:17) +2 γ ( (cid:126)k ) σ + ( (cid:126)k ) ρ S σ − ( (cid:126)k ) . (9)This type of master equation has been studied in Ref. [40],where the environmental electromagnetic field along the x -axis is assumed, A x = E x t . In the following discussion,we lift this restriction and considered the field along the x and y directions. When the quantum-jump term in Eq. (9)can be neglected, the system reduces to a system describedby an effective non-Hermitian Hamiltonian ˆ H eff = d ( (cid:126)k ) σ z − iγ ( (cid:126)k ) | k + (cid:105)(cid:104) k + | . This situation holds valid when we considerthe dynamics in a sufficiently short period of time [41], whichgreatly simplifies the calculation, because it suffices to studythe eigenvectors and eigenvalues of an effective Hamiltonian,instead of performing a time-resolved calculation of the den-sity matrix. After a unitary transformation, we have ˆ H eff = (cid:126)h ( (cid:126)k ) · (cid:126)σ − i Γ ( (cid:126)k ) · σ , (10)where (cid:126)h ( (cid:126)k ) = (cid:126)d ( (cid:126)k ) − ( i/ (cid:126)γ ( (cid:126)k ) . (cid:126) Γ( (cid:126)k ) ≡ (cid:126)γ ( (cid:126)k ) =(Γ x ( (cid:126)k ) , Γ y ( (cid:126)k ) , Γ z ( (cid:126)k )) with Γ x,y,z ( (cid:126)k ) = δβd x,y,z ( (cid:126)k ) and Γ ( (cid:126)k ) = δβd ( (cid:126)k ) . δ = e / (2 π(cid:15) c ) with c standing for thelight speed and β = D + D − . D + D − = (cid:88) α = x,y ( (cid:79) k α d x ( (cid:126)k ) · sin φ − (cid:79) k α d y ( (cid:126)k ) · cos φ ) +( (cid:79) k α d x ( (cid:126)k ) · cos φ · cos θ + (cid:79) k α d y ( (cid:126)k ) · sin φ · cos θ − (cid:79) k α d z ( (cid:126)k ) · sin θ ) . (11)In the following we set δ to be a small constant for con-venience. We find that the energy spectrum of the excitedband is complex and the energy spectrum of ground bandis real: E + = d ( (cid:126)k ) − i Γ( (cid:126)k ) and E − = − d ( (cid:126)k ) , where Γ( (cid:126)k ) = (cid:113)(cid:80) i = x,y,z Γ i ( (cid:126)k ) . The lowest real part of the eigen-spectrum gives the effective ground band, and the imaginarypart of energy is always non-positive that is the decay rate ofthe excited eigenstate [42–45]. It is clear that the decay rate γ ( (cid:126)k ) depends on the Bloch-vector (cid:126)k , which is different fromthe previous studies. This result suggests that the decay rateof the upper Bloch band is different at different positions inmomentum space, leading to interesting features in the Hallconductance.In free space, decay rate can also be expressed as γ ( (cid:126)k ) = π | g ( (cid:126)k ) | ξ ( (cid:126)k ) , where ξ ( (cid:126)k ) = V (2 d ( (cid:126)k )) /π c , (12)is the mode density. Consider the following spectral densityof the environment J c ( ω ) J c ( ω ) = αω ( ωω c ) s − e − ω/ω c , (13)where α is the dimensionless coupling strength, and ω c is thehard upper cutoff. The index s accounts for various physicalsituations, for example for Ohmic spectrum, s = 1 . We have Γ c ( (cid:126)k ) = π J c ( (cid:126)k )= παd ( (cid:126)k ) e − d ( (cid:126)k ) /ω c , (14)where J c ( (cid:126)k ) = (cid:80) n | g ( (cid:126)k ) | δ (2 d ( (cid:126)k ) − ω n ) . III. HALL CONDUCTANCE
In order to derive the response to a constant electric field E , one can introduce a uniform vector potential A ( t ) thatchanges in time such that ∂ t A ( t ) = −E , which can modifythe Bloch vectors, i.e., (cid:126)k ( t ) = (cid:126)k − e A ( t ) . The Hamiltonian ˆ H eff ( (cid:126)k ( t )) satisfies the following time-dependent Schr¨odingerequation. i ∂∂t | u ( (cid:126)k ( t )) (cid:105) = ˆ H eff ( (cid:126)k ( t )) | u ( (cid:126)k ( t )) (cid:105) . (15)Consider a crystal under the perturbation of a weak electricfield. We can write the total Hamiltonian as ˆ H eff ( (cid:126)k ( t )) = ˆ H eff + ˆ H (cid:48) , where ˆ H (cid:48) stands for the perturbation Hamiltonian.If the system is initially in the ground band, it will always stayin this band if the adiabatic condition is met. But now, weconsider the case that there is still a small probability for par-ticles to move from ground to excited band. So we can usethe adiabatic perturbation theory with the perturbation Hamil-tonian ˆ H (cid:48) = − i∂/∂t . The corresponding ground band wavefunction up to the first order in the field strength satisfies [34] | u − (cid:105) = | u − ( (cid:126)k ) (cid:105) − i (cid:104) ˆ u + ( (cid:126)k ) | ∂ t u − ( (cid:126)k ) (cid:105) E − − E + | u + ( (cid:126)k ) (cid:105) , (16)where | u ± ( (cid:126)k ) (cid:105) are the ground band(with ” − ”) and the ex-cited band(with ” + ”) of the Hamiltonian ˆ H eff , respectively. | u n ( (cid:126)k ) (cid:105) and (cid:104) ˆ u n ( (cid:126)k ) | denote the right and the left eigenstatesof ˆ H eff . They satisfy (cid:104) ˆ u m ( k ) | u n ( k ) (cid:105) = δ nm and the com-pleteness relation (cid:88) n | u n ( (cid:126)k ) (cid:105)(cid:104) ˆ u n ( k ) | = (cid:88) n | ˆ u n ( (cid:126)k ) (cid:105)(cid:104) u n ( (cid:126)k ) | = 1 . (17)Similarly, its biorthogonal partner reads | ˆ u − (cid:105) = | ˆ u − ( (cid:126)k ) (cid:105) − i (cid:104) u + ( (cid:126)k ) | ∂ t ˆ u − ( (cid:126)k ) (cid:105) E ∗− − E ∗ + | ˆ u + ( (cid:126)k ) (cid:105) . (18)Next, we use biorthogonal basis vectors to study a dynamicevolution. For any observable ˆ A , the generalised expectationvalues can be defined similarly as (cid:104) ˆ φ | ˆ A | φ (cid:105) [46, 47]. The aver-age velocity ˆ v y in such a state is simplified as ¯ v y = (cid:104) ˆ u − | ˆ v y | u − (cid:105) = (cid:104) ˆ u − ( (cid:126)k ) | ˆ v y | u − ( (cid:126)k ) (cid:105) + ie E x (cid:104) ˆ u − ( (cid:126)k ) | ˆ v y | u + ( (cid:126)k ) (cid:105)(cid:104) ˆ u + ( (cid:126)k ) | ∂∂k x u − ( (cid:126)k ) (cid:105) E − − E + − ie E x (cid:104) ∂∂k x ˆ u − ( (cid:126)k ) | u + ( (cid:126)k ) (cid:105)(cid:104) ˆ u + ( (cid:126)k ) | v y | u − ( (cid:126)k ) (cid:105) E − − E + , (19)where we have assumed that the electric field is along the x -axis. For linear responses, the higher-order terms of E x can be ignored. With the velocity operator defined by ˆ v y = ∂ ˆ H S /∂k y and making use of the following identities (cid:104) ˆ u + ( (cid:126)k ) | ∂∂k x u − ( (cid:126)k ) (cid:105) = (cid:104) ˆ u + ( (cid:126)k ) | ∂ ˆ H eff ∂k x | u − ( (cid:126)k ) (cid:105) E − − E + , (cid:104) ∂∂k x ˆ u − ( (cid:126)k ) | u + ( (cid:126)k ) (cid:105) = (cid:104) ˆ u − ( (cid:126)k ) | ∂ ˆ H eff ∂k x | u + ( (cid:126)k ) (cid:105) E − − E + , (20)we derive the average of v y as follows, ¯ v y = ∂E − ∂k y − eE x h ( (cid:126)k ) ( ∂(cid:126)h ( (cid:126)k ) ∂k x × ∂ (cid:126)d ( (cid:126)k ) ∂k y ) · (cid:126)h ( (cid:126)k ) , (21)where h ( (cid:126)k ) = d ( (cid:126)k ) − i Γ( (cid:126)k ) . For a filled band, the sumover the first term in the velocity is zero. The second termthat contributes to the Hall current can be calculated by (cid:104) ˆ u n ( (cid:126)k ) | σ i | u n ( (cid:126)k ) (cid:105) = nh i ( (cid:126)k ) /h ( (cid:126)k ) . In two-dimensional (cid:126)k space, the Hall current follows, − e (cid:82) dk x dk y / (2 π ) ¯ v y . Theresponse to the external electric field is given by σ H = e π (cid:90) dk x dk y h ( (cid:126)k ) [( ∂(cid:126)h ( (cid:126)k ) ∂k x × ∂(cid:126)h ( (cid:126)k ) ∂k y ) · (cid:126)h ( k )+ i ( ∂(cid:126)h ( (cid:126)k ) ∂k x × ∂(cid:126) Γ( (cid:126)k ) ∂k y ) · (cid:126)h ( (cid:126)k )] . (22)We observe that this response can no longer be proportional tothe Chern number of the non-Hermitian system. The response σ H for the non-Hermitian system can be divided into twoterms. The first term is nothing but the integration of Berrycurvature of the non-Hermitian system. The second term canbe treated as a correction that suggests the relationship be-tween the Chern number and the conductance might fail foropen systems, which is not quantized and a delay in the re-sponse can be found [48]. Specific details would be clearerafter simplifying Eq. (22), lead to σ H =Re( σ H ) + i Im( σ H )= e π (cid:90) dk x dk y P ( (cid:126)k )Ω xy + i e π (cid:90) dk x dk y Q ( (cid:126)k )Ω xy , (23)with P ( (cid:126)k ) = d ( (cid:126)k ) d ( (cid:126)k ) + Γ ( (cid:126)k ) ,Q ( (cid:126)k ) = Γ( (cid:126)k ) d ( (cid:126)k ) d ( (cid:126)k ) + Γ ( (cid:126)k ) , (24)and Ω xy = 12 d ( (cid:126)k ) ( ∂ (cid:126)d ( (cid:126)k ) ∂k x × ∂ (cid:126)d ( (cid:126)k ) ∂k y ) · (cid:126)d ( (cid:126)k ) , (25)which is the key result of this work. Re( σ H ) and Im( σ H ) arethe real and imaginary parts of σ H , respectively. The first part Re( σ H ) is a weighted integration of Berry curvature of theground band, which is no longer quantized in general. Thesecond part Im( σ H ) can be understood as the environment in-duced delay for the system in its response to the electric field,which is reminiscent of the complex admittance in a delaycircuit with capacitor and inductor. So we define the real partof the σ H as the Hall conductance and the imaginary part asthe Hall susceptance. When Γ( (cid:126)k ) = 0 , the Hall conductancereturns to the quantized result and the delay disappears. σ xy = e π (cid:90) dk x dk y Ω xy . (26)Next we present a more detailed analysis of the Hall con-ductance in Eq. (23). First, for the non-Hermitian systems that we studied, the Berry connection A iu − ( (cid:126)k ) for the bandwith energy E − is defined as A iu − ( (cid:126)k ) = i (cid:104) u − ( (cid:126)k ) | ∂∂k i u − ( (cid:126)k ) (cid:105) . (27)With Eq. (27), the Berry curvature follows Ω u − ( (cid:126)k ) = ∇ (cid:126)k × A iu − ( (cid:126)k )= i [ (cid:104) ∂k − ∂k x | ∂k − ∂k y (cid:105) − (cid:104) ∂k − ∂k y | ∂k − ∂k x (cid:105) ]= 12 h ( (cid:126)k ) ( ∂(cid:126)h ( (cid:126)k ) ∂k x × ∂(cid:126)h ( (cid:126)k ) ∂k y ) · (cid:126)h ( (cid:126)k )= 12 d ( (cid:126)k ) ( ∂ (cid:126)d ( (cid:126)k ) ∂k x × ∂ (cid:126)d ( (cid:126)k ) ∂k y ) · (cid:126)d ( (cid:126)k ) . (28)We can prove that non-Hermitian Berry curvature with k -dependent decay rate for the system Ω u − ( (cid:126)k ) is real and equalto the curvature of the Hermitian system Ω xy due to that (cid:126) Γ( (cid:126)k ) and (cid:126)d ( (cid:126)k ) are proportional, which leads the non-HermitIainChern number to be quantized . Meanwhile the P ( (cid:126)k ) termin Eq.(23) depends on Γ( (cid:126)k ) , with P ( (cid:126)k ) → when Γ( (cid:126)k ) → .Therefore in general Re( σ H ) is not quantized and it is nolonger proportional to the Chern number of the non-Hermitiansystem. However, when Γ( (cid:126)k ) → , we can obtain a nearlyquantized Hall conductance. And if the spectral density ofthe environment J ζ ( (cid:126)k ) and d ( (cid:126)k ) are linearly dependent suchthat Γ ζ ( (cid:126)k ) = ζd ( (cid:126)k ) , where ζ is a k -independent constant, wecan get a Hall conductance as Re( σ H ) = [1 / (1 + ζ )] n, n ∈ (0 , ± , i.e., it is not quantized due to the rate [1 / (1 + ζ )] inthe Hall conductance, but it still possesses a platform in thedependence of the conductance on the parameter of the sys-tem.In order to show the validity of the results in Eq.(23), wecompare the above result with the Hall conductance by solv-ing the steady-state solution of the master equation [40, 49]. σ ME = − ie π (cid:90) dk x dk y [ χ ( (cid:126)k ) (cid:104) k + | ∂k − ∂k x (cid:105)(cid:104) ∂k − ∂k y | k + (cid:105) + H . c . ] , (29)where χ ( (cid:126)k ) = d ( (cid:126)k ) / ( d ( (cid:126)k ) − i Γ( (cid:126)k )) . To facilitate the compar-ison with the Hall conductance of the non-Hermitian system,we rewritten the total Hall conductance as σ ME = σ (0)ME + σ (1)ME , (30)where σ (0)ME = ie π (cid:90) dk x dk y P ( (cid:126)k )[ (cid:104) ∂k − ∂k x | ∂k − ∂k y (cid:105) − (cid:104) ∂k − ∂k y | ∂k − ∂k x (cid:105) ] , = e π (cid:90) dk x dk y P ( (cid:126)k )Ω xy ,σ (1)ME = e π (cid:90) dk x dk y Q ( (cid:126)k )[ (cid:104) ∂k − ∂k x | k + (cid:105)(cid:104) k + | ∂k − ∂k y (cid:105) + (cid:104) ∂k − ∂k y | k + (cid:105)(cid:104) k + | ∂k − ∂k x (cid:105) ] , (31)The Hall conductance for open systems by solving the steady-state solution of the master equation consists of two terms.The first term σ (0)ME is the weighted integration of curvature ofthe ground band. The second term σ (1) ME represents the high-order correction of the system-environment coupling to theHall conductance. It can be found that the second term σ (1) ME has a small contribution to Hall conductance and the quantum-jump term has negligible effect on the first-order steady-statesolution of the master equation. The Hall conductance fornon-Hermitian systems Re( σ H ) is then equal to the σ (0)ME ,which verifies that the presented theory is completely validto ignore the quantum-jump term of the master equation forthe system that we studied. IV. EXAMPLE
To demonstrate the response theory for non-Hermitian two-band system, we consider the following example of d α ( (cid:126)k ) d x = sin( k y ) ,d y = − sin( k x ) ,d z = t [2 − m − cos( k x ) − cos( k y )] . (32)This model describes a time reversal symmetry-breaking sys-tem, which might be a magnetic semiconductor with Rashba-type spin-orbit coupling, spin-dependent effective mass, and auniform magnetization in the z direction [50]. This model canbe realized in graphene with F e atoms adsorbed on top, whichpossesses quantum anomalous Hall effect in the presence ofboth Rashba spin-orbit coupling and an exchange field [51].The corresponding complex band structures, E + = d ( (cid:126)k ) − i Γ( (cid:126)k ) and E − = − d ( (cid:126)k ) , are shown in Fig. 1. Consideringthat the mode density is ξ ( (cid:126)k ) , both Re( E ± ) and Im( E ± ) arezero (see Fig. 1 (a,b)) at ( k x , k y ) = (0 , π ) in the tours surfacewith the parameter m = 2 . We conclude that the phase transi-tion point is at m = 2 , the same as in the closed system. Realand imaginary parts of E + and E − for all (cid:126)k with m = 3 (seeFigs. 1 (c,d) have a gap between them and the band structuresare the topologically nontrivial. The imaginary part of the en-ergy eigenvalue E + , the decay rate, depends on the Bloch vec-tor (cid:126)k (see Fig. 1 (b)), which indicates that the relaxation timeof the particle is different at different positions in momentumspace. Comparing Re( E ± ) and Im( E ± ) with the same pa-rameters, we find that their distribution in momentum spaceis similar, which is easy to understand because the relaxationtime of the particle is relatively short at a higher energy Blochband under the influence of the environment.The Hall conductance and Hall susceptance defined inEq. (23) are shown in Fig. 2. In the isolated system limit,i.e., Γ( (cid:126)k ) → , the model Hamiltonian reduces to a Hermi-tian one. In this case, the real part of the Hall conductance Re( σ H ) = − for < m < , while for < m < , Re( σ H ) = 1 , and Im( σ H ) remains zero for closed systems,namely, the Hall conductance is quantized without delay. Forthe open system, however, as shown in fig. 2 (a), we find (a)(c) (b)(d) FIG. 1: Complex band structures of the non-Hermitian Chern insula-tor
Re( E ) ± = ± d ( (cid:126)k ) , | Im( E + ) | = 2Γ( (cid:126)k ) , and | Im( E − ) | = 0 .The mode density ξ ( (cid:126)k ) was taken to plot this figure. (a-b) Realand imaginary parts of the gapless bands with an exceptional pointon ( k x , k y ) = (0 , π ) ) in the tours surface with chosen parameters m = 2 , δ = 0 . , t = 1 . (c-d) Real and imaginary parts of thegapped bands with m = 3 , δ = 0 . , t = 1 . that the phase transition points, i.e., m = 0 , , , remain un-changed, but the Hall conductance is no longer quantized un-der the influence of the environment. Although the Hall con-ductance is nearly quantized with δ = 0 . . The feature ofnon-quantization can be clearly observed when δ is increasedto 0.3 and beyond. Besides, as Eq. (11) shows, the parame-ter β is k − dependent, which leads the Hall conductance tono longer possess a platform in its dependence on the sys-tem parameter. The details of the dependence is closely re-lated to the choice of the model and the spectral density ofthe environment. In fig. 2 (b), we show the Hall susceptance Im( σ H ) which can be understood a delay in the response dueto the system-environment couplings. Im( σ H ) increases with δ ( we show here from 0.1 to 0.3), while Re( σ H ) decreases as δ changing from 0.1 to 0.3.In Fig. 3, we show the Hall conductance of non-Hermitiantwo-band systems Re( σ H ) for different ζ . We consider thespectral density of the environment J ζ ( (cid:126)k ) and d ( (cid:126)k ) are lin-ear. One can find that the Hall conductance still possesses aplatform as the change of the m , but its value is no longer aninteger, for instance, Re( σ H ) = 0 . when ζ = 0 . .In Fig. 4, we show the Hall conductance for non-Hermitiantwo-band systems Re( σ H ) and the Hall conductance for opensystems by solving the master equation σ ME . The same modedensity ξ ( (cid:126)k ) as that used in Fig. 2 (a) has been taken to calcu-late the results in Fig. 4 (a), while Fig. 4 (b) for for the Ohmicspectral density. One can see that the Hall conductance ob-tained by the two methods is almost the same, because thequantum-jump term has negligible effect on the first-ordersteady-state solution of the master equation. Hence in thiscase, it is a good approximation to approximate the model bya non-Hermitian Hamiltonian. The non-Hermitian Hamilto- m R e ( σ H ) ( e / h ) I m ( σ H ) ( e / h ) m m δ =0.1 δ =0.1 δ =0.3 δ =0 δ =0.1 δ =0.3 δ =0 (a)(b) FIG. 2: The Hall conductance
Re( σ H ) and the Hall susceptance Im( σ H ) (in units of e / h ) as a function of m with different δ givenby Eq. (23) and with ξ ( (cid:126)k ) as the mode density. For comparison, thered solid line corresponds to the conventional Hall conductance ofclosed system ( t = 1 ). m R e ( σ H ) ( e / h ) ζ =0.5 ζ =0 FIG. 3: The Hall conductance
Re( σ H ) (in units of e / h ) as a func-tion of m with different ζ given by Eq. (23) with spectral densityof the environment J ζ ( (cid:126)k ) . For comparison, the red solid line cor-responds to the conventional Hall conductance of the correspondingclosed system ( t = 1 ). m (a) σ ( e / h ) Re( σ H ) σ ME m (b) σ ( e / h ) Re( σ H ) σ ME FIG. 4: The Hall conductance
Re( σ H ) and the Hall conductance σ ME (in units of e / h ) as a function of m given by Eq. (23) andEq. (30), respectively. The red solid line is the Hall conductance fornon-Hermitian two-band systems, while the blue dashed line is forthe Hall conductance obtained by solving the master equation. (a)is plotted for mode density ξ ( (cid:126)k ) with δ = 0 . , t = 1 , while (b) isfor the Ohmic spectral density of the environment with α = 0 . , ω c = 1 , t = 1 . nian is obviously more convenient to describe such a system,because it is easy to study the eigenvectors and eigenvaluesof an effective Hamiltonian and the calculation is straightfor-ward. V. CONCLUSIONS
In summary, we have developed the response theory fornon-Hermitian two-band systems and applied it into topolog-ical insulators subjected to environments. An effective non-Hermitian system is obtained by simplifying the Markov mas-ter equation by ignoring the jump terms and a decay rate thatdepends on the Bloch vector is given. Based on this formal-ism, the Hall conductance is calculated by the adiabatic per-turbation theory. We found that although the phase transitionpoint does not changes, the Hall conductance that depends onthe strength of the decay rate and distribution of the electronon the Bloch band is a weighted integration of curvature ofthe ground band and it is not quantized in general. In addi-tion, the system-environment coupling induces a delay in theresponse of the topological insulator to the constant electricfield. Finally, comparing the Hall conductance obtained bythe non-Hermitian two-band model with that by solving themaster equation, we claimed that the non-Hermitian Hamilto-nian description is a good approximation for the open system.
VI. ACKNOWLEDGMENTS
This work was supported by the National Natural ScienceFoundation of China (NSFC) under Grants No. 11775048,No. 11947405, and No. 12047566. [1] M. Z. Hasan, and C. L. Kane, Colloquium: Topological insula-tors, Rev. Mod. Phys. , 3045 (2010), and references therin.[2] B. A. Bernevig, and T. L. Hughes, Topological insulatorsand topeological superconductors (Princeton University Press,Princeton, NJ, 2013).[3] X. L. Qi, and S.-C. Zhang, Topological insulators and super-conductors, Rev. Mod. Phys. , 1057 (2011), and referencestherein.[4] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. denNijs, Quantized Hall conductance in a two-dimensional peri-odic potential, Phys. Rev. Lett. , 405 (1982).[5] K. V. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based onquantized Hall resistance, Phys. Rev. Lett. , 494 (1980).[6] F. D. M. Haldane, Model for a quantum Hall effect with-out landau levels: condensed-matter realization of the ”parityanomaly”, Phys. Rev. Lett. , 2015 (1988).[7] P. San-Jose, J. Cayao, E. Prada, and R. Aguado, Majoranabound states from exceptional points in non-topological super-conductors, Sci. Rep. , 21427 (2016).[8] D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori,Edge modes, degeneracies, and topological numbers in non-Hermitian systems, Phys. Rev. Lett. , 040401 (2017).[9] M. S. Rudner, and L. S. Levitov, Topological transition in a non-Hermitian quantum walk, Phys. Rev. Lett. , 065703 (2009).[10] H. Shen, B. Zhen, and L. Fu, Topological band theory for non-Hermitian Hamiltonians, Phys. Rev. Lett. , 146402 (2018).[11] S. Yao, and Z. Wang, Edge States and topological invariants ofnon-Hermitian systems, Phys. Rev. Lett. , 086803 (2018).[12] S. Yao, F. Song, and Z. Wang, Non-Hermitian Chern bands,Phys. Rev. Lett. , 136802 (2018).[13] Y. Xiong, Why does bulk boundary correspondence fail insome non-Hermitian topological models, J. Phys. Commun. ,035043 (2018).[14] T. E. Lee, Anomalous edge state in a non-Hermitian lattice,Phys. Rev. Lett. , 133903 (2016).[15] K. Kawabata, K. Shiozaki, and M. Ueda, Anomalous helicaledge states in a non-Hermitian Chern insulator, Phys. Rev. B , 165148 (2018).[16] S. Lieu, Topological phases in the non-Hermitian Su-Schrieffer-Heeger model, Phys. Rev. B , 045106 (2018).[17] S. Lieu, Topological symmetry classes for non-Hermitian mod-els and connections to the bosonic BogoliubovCde Gennesequation, Phys. Rev. B , 115135 (2018).[18] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa,and M. Ueda, Topological phases of non-Hermitian systems,Phys. Rev. X , 031079 (2018).[19] K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M.Ueda, Topological unification of time-reversal and particle-holesymmetries in non-Hermitian physics, Nat. Commun. , 297(2019). [20] S.-D. Liang, and G.-Y. Huang, Topological invariance andglobal Berry phase in non-Hermitian systems, Phys. Rev. A ,012118 (2013).[21] T. M. Philip, M. R. Hirsbrunner, and M. J. Gilbert, Loss of Hallconductivity quantization in a non-Hermitian quantum anoma-lous Hall insulator, Phys. Rev. B , 155430 (2018).[22] M. R. Hirsbrunner, T. M. Philip and M. J. Gilbert, Topology andobservables of the non-Hermitian Chern insulator, Phys. Rev. B , 081104(R) (2019).[23] Y. Chen, and H. Zhai, Hall conductance of a non-HermitianChern insulator, Phys. Rev. B , 245130 (2018).[24] L. Pan, X. Chen, Y. Chen and H. Zhai , Non-Hermitian linearresponse theory, Nat. Phys. , 767 (2020).[25] H. Schomerus, Nonreciprocal response theory of non-Hermitian mechanical metamaterials: Response phase transi-tion from the skin effect of zero modes, Phys. Rev. Research ,013058 (2020).[26] L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P.Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H.Obuse, B. C. Sanders, and P. Xue, Observation of topologicaledge states in parity-time-symmetric quantum walks, Nat. Phys. , 1117 (2017).[27] M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J.Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, andM. Khajavikhan, Edge-mode lasing in 1D topological active ar-rays, Phys. Rev. Lett. , 113901 (2018).[28] H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J.D. Joannopoulos, M. Soljaˇci´c, and B. Zhen, Observation of bulkFermi arc and polarization half charge from paired exceptionalpoints, Science , 1009 (2018).[29] M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner,M. Segev, and A. Szameit, Observation of a topological transi-tion in the bulk of a non-Hermitian system, Phys. Rev. Lett. , 040402 (2015).[30] H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, Topological hybrid sil-icon microlasers, Nat. Commun. , 981 (2018).[31] M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, Photoniczero mode in a non-Hermitian photonic lattice, Nat. Commun. , 1308 (2018).[32] S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G.Makris, M. Segev, M. C. Rechtsman, and A. Szameit, Topolog-ically protected bound states in photonic parity-timesymmetriccrystals, Nat. Mater. , 433 (2017).[33] G. Rigolin, G. Ortiz, and V. H. Ponce, Beyond the quantumadiabatic approximation: Adiabatic perturbation theory, Phys.Rev. A , 052508 (2008).[34] A. Bohm, A. Mostafazadeh, H. koizumi, Q. Niu, and J.Zwanziger, The geometric phase in quantum systems (Springer,Berlin, 2003).[35] S. Ib´a˜nez, and J. G. Muga, Adiabaticity condition for non-
Hermitian Hamiltonians, Phys. Rev. A , 033403 (2014).[36] H. M. Weng, R. Yu, X. Hu, X. Dai and Z. Fang, Quantumanomalous Hall effect and related topological electronic states.Adv. Phys. , 227 (2015).[37] W. Q. Zhang, H. Z. Shen, and X. X.Yi, Hall conductance foropen two-band system beyond rotating-wave approximation,Sci. Rep. , 1475 (2018).[38] M. Orszag, Quantum Optics (Springer, Berlin, 2000).[39] H.-P. Breuer, and F. Petruccione,
The Theory of Open QuantumSystems (Oxford University Press, Oxford, 2007).[40] H. Z. Shen, S. S. Zhang, C. M. Dai, and X. X. Yi, Master equa-tion for open two-band systems and its aapplications to Hallconductance, J. Phys. A: Math. Theor. , 065302 (2018).[41] S. D¨urr, J. J. Garcia-Ripoll, N. Syassen, D. M. Bauer, M.Lettner, J. I. Cirac, and G. Rempe, Lieb-Liniger model ofa dissipation-induced Tonks-Girardeau gas, Phys. Rev. A ,023614 (2009).[42] T. E. Lee and C.-K. Chan, Heralded Magnetism in Non-Hermitian Atomic Systems, Phys. Rev. X , 041001 (2014).[43] Y. Ashida, S. Furukawa, and M. Ueda, Quantum critical behav-ior influenced by measurement backaction in ultracold gases,Phys. Rev. A , 053615 (2016).[44] Y. Ashida, S. Furukawa, and M. Ueda, Parity-time-symmetric quantum critical phenomena, Nat. Commun. , 15791 (2017).[45] K. Yamamoto, M. Nakagawa, K. Adachi, K. Takasan, M. Ueda,and N. Kawakami, Theory of non-Hermitian fermionic super-fluidity with a complex-valued interaction, Phys. Rev. Lett. ,123601 (2019).[46] P Ziesche, Generalisation of the Hellmann-Feynman theoremto Gamow states, J. Phys. A: Math. Gen. , 2859 (1987).[47] G. Dattoli, and A. Torre, non-Hermitian evolution of two-levelquantum systems, Phys. Rev. A , 1467 (1990).[48] D. X. Li, H. Z. Shen, H. D. Liu, and X. X. Yi, Effect of spinrelaxations on the spin mixing conductances for a bilayer struc-ture, Sci. Rep. , 1475 (2018).[49] H. Z. Shen, W. Wang, and X. X. Yi, Hall conductance and topo-logical invariant for open systems, Sci. Rep. , 1038 (2014).[50] X. L. Qi, Y. S. Wu, and S.-C. Zhang, Topological quantizationof the spin Hall effect in two-dimensional paramagnetic semi-conductors, Phys. Rev. B , 085308 (2006).[51] Z. Qiao, S. A. Yang, W. Feng, W.-K. Tse, J. Ding, Y. Yao,J. Wang, Q. Niu, Quantum anomalous Hall effect in graphenefrom Rashba and exchange effects, Phys, Rev B82